I used to ride horses, a lot. I found it to be a really wonderful pastime, and when I'm too old to make an mtb go uphill anymore I'm considering renewing my horseriding skills.
Horses can be trained to be in traffic- the Met Police and other horses can and do do some amazing things in crowds, disturbances and riots. However, not all horse owners/ riders can manage to get their horses to do the same, much as not all MTB riders can do amazing DH rides, tricks and stunts etc.
As I understand it however, bicycles and horses both have a right to use the public highway (rights of way are public highways I believe) and AFAIK in both cases the rider has to be 'in control' of their horse/ bike/ means of conveyance/ vehicle.
Over the years I have ridden/ owned a number of horses, much as I have with my bikes. I had an emotional bond with the horses, like with my dog at present and tbh the bond was stronger than the one I had/ have with my bikes.
One horse in particular was VERY easily spooked by traffic. I chose never to ride him on busy roads. I was not skilled enough to control him, so I avoided the problem. I personally believe that this is what horse riders should all do. If you cannot control your 'ride' in all reasonably predictable potential situations then you need to adapt in some way or another.
I learnt this the hard way. Phoenix, my horse spooked when my brother was unable to control him. He was hit by an articulated truck and desperately injured. My brother managed to jump clear just in time- Goodness knows how, and Phoenix was hit by a 44t lorry. It was not a good day for anybody.
Like cyclists need to be in control, horse riders need to be able to be in total control of their horses when they are on the public highway, the law requires it and if a person is not in control then they will become liable for any outcome. Cyclists perhaps should also be aware that horses not are always nice, and that a kick from a horse will 'kin hurt! so don't get too close.
However we all have a right to use the bridleways.